Quasi perfect codes in the cartesian product of some graphs
S. A. Mane, N. V. Shinde
TL;DR
The paper develops a general method to construct quasi-perfect $e$-error-correcting codes in Cartesian products of graphs by leveraging existing perfect $e$-error-correcting codes in a base graph $G$. The authors provide explicit constructions for $G \square P_n$ and $G \square C_n$, and extend these ideas to higher-dimensional products such as $P_m \square P_n \square P_{6k-2}$ and $C_m \square C_n \ square C_{6k}$, including a $P_4 \square P_4 \square P_4$ example derived from a smaller dimension. They further develop quasi-perfect codes in $C_n \square C_n \square C_l$ for $e \le 2$ based on $C_n \square C_n$ codes, with concrete results for $3 \le n \le 19$ and various $l$, employing tiling and layering techniques. In the path-product setting, they identify specific parameter regimes for $P_m \square P_n$ and $P_m \square P_n \square P_l$ that admit quasi-perfect codes, and present multiple explicit code constructions achieving the desired minimum distance and covering radius. Overall, the work broadens the catalog of quasi-perfect codes in graph products and provides systematic methods to derive new codes from existing perfect codes, with potential implications for network design and graph-based coding theory.
Abstract
An important question in the study of quasi-perfect codes is whether such codes can be constructed for all possible lengths $n$. In this paper, we address this question for specific values of $n$. First, we investigate the existence of quasi-perfect codes in the Cartesian product of a graph $G$ and a path (or cycle), assuming that $G$ admits a perfect code. Second, we explore quasi-perfect codes in the Cartesian products of two or three cycles, $C_m\square C_n$ and $C_m\square C_n\square C_l$, as well as in the Cartesian products of two or three paths, $P_m\square P_n$ and $P_m\square P_n\square P_l$.